Mathematical Analysis of the Water Wave Problem

水波的数学问题涉及在重力的影响下，将无粘性、不可压缩和无旋流体与零密度区域（即空气）分离的界面的运动。假设表面张力为零。在以前的工作中，PI建立了充分正则的Sobolev空间中水波问题的局部时间适定性。这个项目的重点是对非线性效应的理解，海底地形对表面波运动的长时间行为的影响，并找到一些适当的功能空间设置描述不那么光滑的水波运动。水波是我们日常生活中最熟悉的体验之一。诸如流氓波和波浪破碎等现象的原因仍然是积极研究的主题。了解它们对海洋生物和海洋学具有重要意义。这个项目的目的是提供一些定性的了解水波运动。

The mathematical problem of water wave concerns the motion of the interface separating an inviscid, incompressible and irrotational fluid, under the influence of gravity, from a region of zero density (i.e. air). It is assumed that the surface tension is zero. In previous works, the PI established the local in time well-posedness of the water wave problem in Sobolev spaces of sufficient regularity. The focuses of this project are on the understanding of the nonlinear effect, the effect of the bottom topography to the long time behavior of the surface wave motions, and on finding some appropriate functional space settings describing not so smooth water wave motions. Water waves are one of our most familiar experiences in daily life. The causes of phenomena such as the rogue waves and wave breaking are still subjects of active research. Their understanding is of great importance to marine life and oceanography. This project aims at providing some qualitative understanding of water wave motion.